In the absence of external torque for a body revolving about any axis, the quantity that remains constant is

If the ice at the poles of the Earth melts and flows towards the equator, will the angular velocity of the Earth increase or decrease?

$A$ man stands on a rotating platform with his arms stretched horizontally,holding a $5\; kg$ weight in each hand. The angular speed of the platform is $30$ revolutions per minute. The man then brings his arms close to his body,with the distance of each weight from the axis changing from $90\; cm$ to $20\; cm$. The moment of inertia of the man together with the platform may be taken to be constant and equal to $7.6\; kg\; m^2$.
$(a)$ What is his new angular speed? (Neglect friction.)
$(b)$ Is kinetic energy conserved in the process? If not,from where does the change come about?

$A$ circular platform is mounted on a frictionless vertical axle. Its radius $R = 2\, m$ and its moment of inertia about the axle is $200\, kg\, m^2$. It is initially at rest. $A$ $50\, kg$ man stands on the edge of the platform and begins to walk along the edge at the speed of $1\, m/s$ relative to the ground. Time taken by the man to complete one revolution is

$A$ circular platform of moment of inertia $100 \ kg\ m^2$ with a boy of mass $20 \ kg$ standing at the centre completes one rotation in $10 \ s$. If he walks $2 \ m$ away from the centre along the radius,then one rotation (with respect to the ground) will be completed in $.... \ s$.

$A$ thin circular disc of mass $M$ and radius $R$ is rotating in a horizontal plane with an angular velocity $\omega$ about an axis passing through its center and perpendicular to its plane. If another disc of the same radius but one-fourth of its mass is placed gently on top of it,what will be the angular velocity of the new system?